Tampered random variable modeling for multiple step-stress life test

Abstract

In this paper, we introduce the tampered random variable (TRV) modeling in multiple step-stress life testing experiments. Here be prespecified time points and be k prefixed stress levels with si being the stress level in force during the time interval for with and We define the tampered random variable in multiple step-stress scenario and calculate the PDF, CDF, and Hazard rate for the proposed tampered variable We derive a general expression for the expectation of under different number k of stress levels and also obtain some results on stochastic ordering for different k. All these results are obtained under arbitrary baseline (under normal stress condition with stress level s 1) life distribution. In particular, we consider exponential distribution with mean θ and Weibull distribution with scale parameter λ and shape parameter α for specific expressions. We also prove some results on equivalence of the TRV modeling with the two other existing models for step-stress life testing, namely, cumulative exposure and tampered failure rate. Finally, we consider some variations of the modeling approach for to include incorporation of the stress levels, discrete life time, bivariate or multivariate life times.